Magnetic resonance imaging (MRI) is a medical diagnostic imaging technique used to diagnose many types of medical conditions. An MRI system includes a main magnet for generating a main magnetic field through an examination region. The main magnet is arranged such that its geometry defines the examination region. The main magnetic field causes the magnetic moments of a small majority of the various nuclei within the body to be aligned in a parallel or anti-parallel arrangement. The aligned magnetic moments rotate around the equilibrium axis with a frequency that is characteristic for the nuclei to be imaged. An external radiofrequency (RF) field applied by other hardware within the MRI system perturbs the magnetization from its equilibrium state. Upon termination of the application of the RF pulse, the magnetization relaxes to its initial state. During relaxation the time varying magnetic moment induces a detectable time varying voltage in the receive coil. The time varying voltage can be detected by the receive mode of the transmit coil itself, or by an independent receive only coil.
MRI systems are made of many hardware components that work in conjunction with specialized software to produce the final images FIG. 1 illustrates an MRI system with the front cover removed so the main hardware components can be seen. Magnet 12 is the main hardware component of MRI system 10 and is responsible for producing the uniform main magnetic field, B0. Magnets used in MRI systems can produce either a horizontal or a vertical magnetic field.
Within the volume defined by main magnet 12 there is at least one gradient coil 14. Gradient coils 14 produces substantially linear spatially varying magnetic fields within the imaging volume that are coincidental with the direction of the main magnetic field but vary along the three orthogonal directions (x, y, z) of the Cartesian coordinate system. Initially, gradient coils included only single primary gradient coil unit 18, however, the presence of spurious and spatially dependent eddy currents on the magnet's Dewar structure necessitated the need to shield the gradient magnetic field at the magnet's vicinity. Secondary shielding gradient coil 20 is added to the primary gradient coil's structure with a required minimum gap between the primary and secondary gradient coil structures.
Self-shielded gradient coils of prior art are usually designed using the following general steps:                1. The shield coil is assumed to be much longer than the primary coil;        2. The continuous current densities on both the primary and shield coil are derived to deliver the desired field quality characteristics of the gradient coil;        3. The continuous current density on the shield coil is truncated to match the desired length of the system;        4. The continuous current densities are discretized using the standard techniques, described below; and        5. The discretized solution is verified to ensure the solution satisfies the required characteristics.        
One of the problems in past methods of gradient coil design is the discretization procedure that approximates the continuous current densities by a number of current carrying conductors (step 4 of the above method). The shape of the current carrying conductors determines the electromagnetic properties of the gradient coil. These include the gradient field strength, field quality characteristics inside the Field-of-View (FoV), linearity and uniformity along the principal axis and orthogonal axes respectively, leakage of the magnetic field into the region where eddy currents might be induced, coil resistance, slew rate, etc. The output of the discretization procedure determines the positions of the current carrying conductors and their minimum widths, which govern the coil resistance, thus the power dissipated by the coil and the possible slew rate. For a self-shielded gradient coil the level of the eddy currents is sensitive to discretization procedure When designing the gradient coil it is also important to be able to minimize the net thrust Lorentz force exerted on the coil due to the presence of the main magnet field. This force is sensitive to the position of the current carrying conductors, especially near the end of the coil.
The common practice of previous gradient coil designs is to derive the continuous current densities that provide the desired characteristics of the gradient coil. It is sometimes difficult to find a continuous solution that satisfies the field quality characteristics within the FoV and where the net thrust force is nullified by constraint. Depending on the magnet configuration the nullification of the thrust force can lead to an energy penalty or even to reversed current patterns on either the primary coil or the shield coil, which in fact leads to the energy/inductance penalty.
Another problem that arises in previous methods of gradient coil design is that eddy currents are induced in the worm bore or the cold shield of the magnet. Eddy currents in the cold shield result in distortions of the images. Because of the proximity of the cold shield to the magnet's coils the forces on the cold shield could be significant. These significant forces on the cold shield may lead to the vibration of the cold shield and even quenching of the magnet.
In previous gradient coil design methods the continuous current solution that satisfies the field quality characteristics within the FoV is obtained using either minimum inductance or target field method of Turner. In the design of an axial gradient coil, the continuous current densities and available power supply determine the minimum number of turns on both the primary and shield coil according to the method of Turner:
                                                        ∫              0                              L                P                                      ⁢                                                            f                  φ                                      (                    P                    )                                                  ⁡                                  (                  z                  )                                            ⁢                                                          ⁢                              ⅆ                z                                              =                                    N              P                        ⁢                          I              P                                      ,                                            ∫              0                              L                S                                      ⁢                                                            f                  φ                                      (                    S                    )                                                  ⁡                                  (                  z                  )                                            ⁢                                                          ⁢                              ⅆ                z                                              =                                    -                              N                S                                      ⁢                          I              S                                                          (                  Equation          ⁢                                          ⁢          1                )            In this two-part equation, where the first portion of the Equation deals with the primary coil and the second portion deals with the shield coil, fφ(P,S)(z) is the φ-components of the continuous current density on primary coil (P) and shield coil (S), LP,S is the half-length of the primary/shield coil, NP,S, IP,S are the number of turns and the current on half of the primary/shield coil and the current in the primary/shield, respectively. Ideally the currents on the primary and the shield coil should be equal to each other. In practice the latter is difficult to achieve (IP≠IS) because of the free parameters in the problem: the geometry of the coil is determined by the space available and usually is at premium.
The positions of the current centroids (commonly called hoops) Zn,m(P,S) are usually determined by the integrated current density according to the following equation
                                                                        ∫                0                                  Z                  n                                      (                    P                    )                                                              ⁢                                                                    f                    φ                                          (                      P                      )                                                        ⁡                                      (                    z                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  z                                                      -                                          (                                  n                  -                  0.5                                )                            ⁢                              I                P                                              =          0                ,                                  ⁢                                                            ∫                0                                  Z                  m                                      (                    S                    )                                                              ⁢                                                                    f                    φ                                          (                      S                      )                                                        ⁡                                      (                    z                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  z                                                      +                                          (                                  m                  -                  0.5                                )                            ⁢                              I                S                                              =          0                                    (                  Equation          ⁢                                          ⁢          2                )            Within Equation 2, n is an integer between 1 and the number of turns on the primary coil, and m is an integer between 1 and the number of turns on the secondary coil (1≦n≦NP, 1≦m≦NS).
In designing a transverse gradient using previous methods, the current paths on the primary/shield coil are defined by the following equation:
                                                        φ              =                                                Φ                  n                                ⁡                                  (                  z                  )                                                                                                                                          Φ                  n                                ⁡                                  (                  z                  )                                            =                              arc                ⁢                                                                  ⁢                                  cos                  ⁡                                      (                                                                  S                        n                                            /                                                                        f                          z                                                ⁡                                                  (                          z                          )                                                                                      )                                                                                                                                          S                n                            =                                                                    n                    -                    0.5                                    N                                ⁢                                                      f                    z                                    ⁡                                      (                                          z                      eye                                        )                                                                                                          (                  Equation          ⁢                                          ⁢          3                )            Here fz(z) is the z-component of the continuous current density, Zeye is the so called eye-position of the coil where the φ-component of the continuous current density is equal zero:
                    f        φ            ⁡              (                  z          eye                )              =                  0        :                              f            z                    ⁡                      (                          z              eye                        )                              =                        max                      z            ∈                          [                              0                ,                L                            ]                                      ⁢                  {                                    f              z                        ⁡                          (              z              )                                }                      ,N is the number of loops in one quadrant of the coil, and n is an integer: 1≦n≦N.
At φ=0 Eq. (3) determines the positions of Z-intercepts: the positions where the current paths intercept the cardinal axis as it is shown in FIG. 2. In FIG. 2 N-initial and N-final Z-intercepts are shown. Information about a transverse gradient can be expressed through the Z-intercepts and the current paths.